Properties

Label 2.88.a.b
Level 2
Weight 88
Character orbit 2.a
Self dual yes
Analytic conductor 95.867
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 88 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(95.8667262922\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 17\cdot 29 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8796093022208 q^{2} +(\)\(10\!\cdots\!32\)\( - \beta_{1}) q^{3} +\)\(77\!\cdots\!64\)\( q^{4} +(-\)\(10\!\cdots\!90\)\( - 1822922575 \beta_{1} + \beta_{2}) q^{5} +(-\)\(88\!\cdots\!56\)\( + 8796093022208 \beta_{1}) q^{6} +(\)\(21\!\cdots\!76\)\( + 1846032815820606 \beta_{1} + 714006 \beta_{2} + 4213 \beta_{3}) q^{7} -\)\(68\!\cdots\!12\)\( q^{8} +(\)\(20\!\cdots\!37\)\( - 70457522433696719338 \beta_{1} - 19291700786 \beta_{2} + 197927972 \beta_{3}) q^{9} +O(q^{10})\) \( q -8796093022208 q^{2} +(\)\(10\!\cdots\!32\)\( - \beta_{1}) q^{3} +\)\(77\!\cdots\!64\)\( q^{4} +(-\)\(10\!\cdots\!90\)\( - 1822922575 \beta_{1} + \beta_{2}) q^{5} +(-\)\(88\!\cdots\!56\)\( + 8796093022208 \beta_{1}) q^{6} +(\)\(21\!\cdots\!76\)\( + 1846032815820606 \beta_{1} + 714006 \beta_{2} + 4213 \beta_{3}) q^{7} -\)\(68\!\cdots\!12\)\( q^{8} +(\)\(20\!\cdots\!37\)\( - 70457522433696719338 \beta_{1} - 19291700786 \beta_{2} + 197927972 \beta_{3}) q^{9} +(\)\(88\!\cdots\!20\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} - 8796093022208 \beta_{2}) q^{10} +(\)\(74\!\cdots\!12\)\( + \)\(61\!\cdots\!05\)\( \beta_{1} - 111812372519476 \beta_{2} + 1078925628602 \beta_{3}) q^{11} +(\)\(77\!\cdots\!48\)\( - \)\(77\!\cdots\!64\)\( \beta_{1}) q^{12} +(-\)\(44\!\cdots\!78\)\( + \)\(24\!\cdots\!97\)\( \beta_{1} + 448615718005475761 \beta_{2} + 1937036307534328 \beta_{3}) q^{13} +(-\)\(18\!\cdots\!08\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - 6280463194414645248 \beta_{2} - 37057939902562304 \beta_{3}) q^{14} +(\)\(84\!\cdots\!20\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} + \)\(29\!\cdots\!62\)\( \beta_{2} + 1720528777181671875 \beta_{3}) q^{15} +\)\(59\!\cdots\!96\)\( q^{16} +(-\)\(17\!\cdots\!54\)\( + \)\(40\!\cdots\!90\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} - 82346674465175417836 \beta_{3}) q^{17} +(-\)\(18\!\cdots\!96\)\( + \)\(61\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3}) q^{18} +(\)\(95\!\cdots\!40\)\( - \)\(10\!\cdots\!13\)\( \beta_{1} + \)\(86\!\cdots\!84\)\( \beta_{2} + \)\(15\!\cdots\!82\)\( \beta_{3}) q^{19} +(-\)\(78\!\cdots\!60\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(77\!\cdots\!64\)\( \beta_{2}) q^{20} +(-\)\(74\!\cdots\!68\)\( - \)\(49\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} + \)\(62\!\cdots\!32\)\( \beta_{3}) q^{21} +(-\)\(65\!\cdots\!96\)\( - \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(98\!\cdots\!08\)\( \beta_{2} - \)\(94\!\cdots\!16\)\( \beta_{3}) q^{22} +(\)\(66\!\cdots\!32\)\( + \)\(82\!\cdots\!70\)\( \beta_{1} + \)\(43\!\cdots\!74\)\( \beta_{2} + \)\(18\!\cdots\!27\)\( \beta_{3}) q^{23} +(-\)\(68\!\cdots\!84\)\( + \)\(68\!\cdots\!12\)\( \beta_{1}) q^{24} +(\)\(67\!\cdots\!75\)\( + \)\(96\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3}) q^{25} +(\)\(38\!\cdots\!24\)\( - \)\(21\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!88\)\( \beta_{2} - \)\(17\!\cdots\!24\)\( \beta_{3}) q^{26} +(\)\(24\!\cdots\!00\)\( - \)\(28\!\cdots\!82\)\( \beta_{1} + \)\(40\!\cdots\!24\)\( \beta_{2} - \)\(11\!\cdots\!98\)\( \beta_{3}) q^{27} +(\)\(16\!\cdots\!64\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} + \)\(55\!\cdots\!84\)\( \beta_{2} + \)\(32\!\cdots\!32\)\( \beta_{3}) q^{28} +(-\)\(23\!\cdots\!10\)\( - \)\(10\!\cdots\!47\)\( \beta_{1} + \)\(71\!\cdots\!09\)\( \beta_{2} + \)\(82\!\cdots\!32\)\( \beta_{3}) q^{29} +(-\)\(74\!\cdots\!60\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(25\!\cdots\!96\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3}) q^{30} +(\)\(29\!\cdots\!92\)\( - \)\(52\!\cdots\!36\)\( \beta_{1} - \)\(16\!\cdots\!96\)\( \beta_{2} + \)\(47\!\cdots\!92\)\( \beta_{3}) q^{31} -\)\(52\!\cdots\!68\)\( q^{32} +(-\)\(24\!\cdots\!16\)\( - \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(23\!\cdots\!02\)\( \beta_{2} - \)\(26\!\cdots\!04\)\( \beta_{3}) q^{33} +(\)\(15\!\cdots\!32\)\( - \)\(35\!\cdots\!20\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} + \)\(72\!\cdots\!88\)\( \beta_{3}) q^{34} +(\)\(35\!\cdots\!60\)\( - \)\(35\!\cdots\!00\)\( \beta_{1} + \)\(33\!\cdots\!16\)\( \beta_{2} - \)\(40\!\cdots\!00\)\( \beta_{3}) q^{35} +(\)\(15\!\cdots\!68\)\( - \)\(54\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(15\!\cdots\!08\)\( \beta_{3}) q^{36} +(\)\(45\!\cdots\!66\)\( - \)\(82\!\cdots\!55\)\( \beta_{1} - \)\(23\!\cdots\!67\)\( \beta_{2} + \)\(33\!\cdots\!84\)\( \beta_{3}) q^{37} +(-\)\(83\!\cdots\!20\)\( + \)\(88\!\cdots\!04\)\( \beta_{1} - \)\(76\!\cdots\!72\)\( \beta_{2} - \)\(13\!\cdots\!56\)\( \beta_{3}) q^{38} +(-\)\(12\!\cdots\!96\)\( - \)\(83\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!50\)\( \beta_{2} + \)\(14\!\cdots\!75\)\( \beta_{3}) q^{39} +(\)\(68\!\cdots\!80\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(68\!\cdots\!12\)\( \beta_{2}) q^{40} +(\)\(13\!\cdots\!02\)\( + \)\(20\!\cdots\!64\)\( \beta_{1} + \)\(16\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!84\)\( \beta_{3}) q^{41} +(\)\(65\!\cdots\!44\)\( + \)\(43\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(54\!\cdots\!56\)\( \beta_{3}) q^{42} +(-\)\(39\!\cdots\!88\)\( + \)\(71\!\cdots\!13\)\( \beta_{1} + \)\(51\!\cdots\!04\)\( \beta_{2} - \)\(32\!\cdots\!08\)\( \beta_{3}) q^{43} +(\)\(57\!\cdots\!68\)\( + \)\(47\!\cdots\!20\)\( \beta_{1} - \)\(86\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!28\)\( \beta_{3}) q^{44} +(-\)\(34\!\cdots\!30\)\( - \)\(14\!\cdots\!75\)\( \beta_{1} + \)\(21\!\cdots\!57\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{45} +(-\)\(58\!\cdots\!56\)\( - \)\(72\!\cdots\!60\)\( \beta_{1} - \)\(38\!\cdots\!92\)\( \beta_{2} - \)\(16\!\cdots\!16\)\( \beta_{3}) q^{46} +(-\)\(29\!\cdots\!84\)\( - \)\(51\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(16\!\cdots\!94\)\( \beta_{3}) q^{47} +(\)\(60\!\cdots\!72\)\( - \)\(59\!\cdots\!96\)\( \beta_{1}) q^{48} +(\)\(11\!\cdots\!33\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} + \)\(19\!\cdots\!24\)\( \beta_{3}) q^{49} +(-\)\(59\!\cdots\!00\)\( - \)\(84\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3}) q^{50} +(-\)\(22\!\cdots\!28\)\( + \)\(22\!\cdots\!46\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(83\!\cdots\!38\)\( \beta_{3}) q^{51} +(-\)\(34\!\cdots\!92\)\( + \)\(18\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3}) q^{52} +(-\)\(80\!\cdots\!78\)\( + \)\(96\!\cdots\!21\)\( \beta_{1} - \)\(97\!\cdots\!23\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3}) q^{53} +(-\)\(21\!\cdots\!00\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(35\!\cdots\!92\)\( \beta_{2} + \)\(10\!\cdots\!84\)\( \beta_{3}) q^{54} +(-\)\(25\!\cdots\!80\)\( - \)\(86\!\cdots\!50\)\( \beta_{1} + \)\(63\!\cdots\!42\)\( \beta_{2} - \)\(10\!\cdots\!75\)\( \beta_{3}) q^{55} +(-\)\(14\!\cdots\!12\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(48\!\cdots\!72\)\( \beta_{2} - \)\(28\!\cdots\!56\)\( \beta_{3}) q^{56} +(\)\(61\!\cdots\!80\)\( - \)\(16\!\cdots\!42\)\( \beta_{1} + \)\(63\!\cdots\!34\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3}) q^{57} +(\)\(20\!\cdots\!80\)\( + \)\(90\!\cdots\!76\)\( \beta_{1} - \)\(63\!\cdots\!72\)\( \beta_{2} - \)\(72\!\cdots\!56\)\( \beta_{3}) q^{58} +(-\)\(20\!\cdots\!20\)\( + \)\(84\!\cdots\!01\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(46\!\cdots\!76\)\( \beta_{3}) q^{59} +(\)\(65\!\cdots\!80\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3}) q^{60} +(\)\(22\!\cdots\!82\)\( + \)\(28\!\cdots\!93\)\( \beta_{1} - \)\(58\!\cdots\!67\)\( \beta_{2} - \)\(12\!\cdots\!16\)\( \beta_{3}) q^{61} +(-\)\(26\!\cdots\!36\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!68\)\( \beta_{2} - \)\(42\!\cdots\!36\)\( \beta_{3}) q^{62} +(\)\(17\!\cdots\!12\)\( + \)\(35\!\cdots\!86\)\( \beta_{1} + \)\(25\!\cdots\!14\)\( \beta_{2} + \)\(13\!\cdots\!97\)\( \beta_{3}) q^{63} +\)\(46\!\cdots\!44\)\( q^{64} +(\)\(28\!\cdots\!20\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2} - \)\(46\!\cdots\!00\)\( \beta_{3}) q^{65} +(\)\(21\!\cdots\!28\)\( + \)\(13\!\cdots\!68\)\( \beta_{1} - \)\(20\!\cdots\!16\)\( \beta_{2} + \)\(23\!\cdots\!32\)\( \beta_{3}) q^{66} +(-\)\(16\!\cdots\!24\)\( - \)\(15\!\cdots\!21\)\( \beta_{1} + \)\(75\!\cdots\!72\)\( \beta_{2} + \)\(99\!\cdots\!06\)\( \beta_{3}) q^{67} +(-\)\(13\!\cdots\!56\)\( + \)\(31\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} - \)\(63\!\cdots\!04\)\( \beta_{3}) q^{68} +(-\)\(36\!\cdots\!76\)\( - \)\(82\!\cdots\!76\)\( \beta_{1} + \)\(62\!\cdots\!92\)\( \beta_{2} - \)\(15\!\cdots\!84\)\( \beta_{3}) q^{69} +(-\)\(31\!\cdots\!80\)\( + \)\(31\!\cdots\!00\)\( \beta_{1} - \)\(29\!\cdots\!28\)\( \beta_{2} + \)\(35\!\cdots\!00\)\( \beta_{3}) q^{70} +(-\)\(45\!\cdots\!48\)\( - \)\(18\!\cdots\!78\)\( \beta_{1} - \)\(58\!\cdots\!86\)\( \beta_{2} - \)\(74\!\cdots\!03\)\( \beta_{3}) q^{71} +(-\)\(13\!\cdots\!44\)\( + \)\(47\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!64\)\( \beta_{3}) q^{72} +(\)\(88\!\cdots\!22\)\( + \)\(21\!\cdots\!66\)\( \beta_{1} - \)\(76\!\cdots\!38\)\( \beta_{2} + \)\(77\!\cdots\!76\)\( \beta_{3}) q^{73} +(-\)\(40\!\cdots\!28\)\( + \)\(72\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3}) q^{74} +(\)\(17\!\cdots\!00\)\( - \)\(92\!\cdots\!75\)\( \beta_{1} - \)\(34\!\cdots\!20\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(73\!\cdots\!60\)\( - \)\(78\!\cdots\!32\)\( \beta_{1} + \)\(66\!\cdots\!76\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{3}) q^{76} +(\)\(98\!\cdots\!12\)\( + \)\(65\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} + \)\(57\!\cdots\!04\)\( \beta_{3}) q^{77} +(\)\(11\!\cdots\!68\)\( + \)\(73\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3}) q^{78} +(\)\(39\!\cdots\!80\)\( + \)\(49\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!88\)\( \beta_{2} - \)\(82\!\cdots\!26\)\( \beta_{3}) q^{79} +(-\)\(60\!\cdots\!40\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(59\!\cdots\!96\)\( \beta_{2}) q^{80} +(-\)\(49\!\cdots\!19\)\( + \)\(22\!\cdots\!50\)\( \beta_{1} + \)\(16\!\cdots\!02\)\( \beta_{2} - \)\(35\!\cdots\!04\)\( \beta_{3}) q^{81} +(-\)\(12\!\cdots\!16\)\( - \)\(17\!\cdots\!12\)\( \beta_{1} - \)\(14\!\cdots\!64\)\( \beta_{2} - \)\(26\!\cdots\!72\)\( \beta_{3}) q^{82} +(\)\(27\!\cdots\!92\)\( + \)\(13\!\cdots\!63\)\( \beta_{1} - \)\(31\!\cdots\!68\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3}) q^{83} +(-\)\(57\!\cdots\!52\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!76\)\( \beta_{2} + \)\(48\!\cdots\!48\)\( \beta_{3}) q^{84} +(-\)\(22\!\cdots\!40\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!94\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3}) q^{85} +(\)\(34\!\cdots\!04\)\( - \)\(62\!\cdots\!04\)\( \beta_{1} - \)\(45\!\cdots\!32\)\( \beta_{2} + \)\(28\!\cdots\!64\)\( \beta_{3}) q^{86} +(\)\(50\!\cdots\!80\)\( - \)\(45\!\cdots\!46\)\( \beta_{1} + \)\(41\!\cdots\!10\)\( \beta_{2} + \)\(11\!\cdots\!55\)\( \beta_{3}) q^{87} +(-\)\(50\!\cdots\!44\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(76\!\cdots\!12\)\( \beta_{2} - \)\(73\!\cdots\!24\)\( \beta_{3}) q^{88} +(-\)\(27\!\cdots\!90\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} + \)\(13\!\cdots\!50\)\( \beta_{2} - \)\(57\!\cdots\!00\)\( \beta_{3}) q^{89} +(\)\(30\!\cdots\!40\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3}) q^{90} +(\)\(19\!\cdots\!72\)\( - \)\(94\!\cdots\!00\)\( \beta_{1} - \)\(38\!\cdots\!88\)\( \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3}) q^{91} +(\)\(51\!\cdots\!48\)\( + \)\(63\!\cdots\!80\)\( \beta_{1} + \)\(33\!\cdots\!36\)\( \beta_{2} + \)\(14\!\cdots\!28\)\( \beta_{3}) q^{92} +(\)\(30\!\cdots\!44\)\( - \)\(68\!\cdots\!52\)\( \beta_{1} + \)\(51\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!32\)\( \beta_{3}) q^{93} +(\)\(25\!\cdots\!72\)\( + \)\(45\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{94} +(\)\(90\!\cdots\!00\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(17\!\cdots\!90\)\( \beta_{2} + \)\(10\!\cdots\!75\)\( \beta_{3}) q^{95} +(-\)\(53\!\cdots\!76\)\( + \)\(52\!\cdots\!68\)\( \beta_{1}) q^{96} +(\)\(21\!\cdots\!46\)\( + \)\(30\!\cdots\!58\)\( \beta_{1} - \)\(46\!\cdots\!78\)\( \beta_{2} - \)\(25\!\cdots\!44\)\( \beta_{3}) q^{97} +(-\)\(10\!\cdots\!64\)\( + \)\(62\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!96\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3}) q^{98} +(\)\(55\!\cdots\!44\)\( + \)\(37\!\cdots\!33\)\( \beta_{1} + \)\(18\!\cdots\!12\)\( \beta_{2} + \)\(28\!\cdots\!76\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 35184372088832q^{2} + \)\(40\!\cdots\!28\)\(q^{3} + \)\(30\!\cdots\!56\)\(q^{4} - \)\(40\!\cdots\!60\)\(q^{5} - \)\(35\!\cdots\!24\)\(q^{6} + \)\(85\!\cdots\!04\)\(q^{7} - \)\(27\!\cdots\!48\)\(q^{8} + \)\(82\!\cdots\!48\)\(q^{9} + O(q^{10}) \) \( 4q - 35184372088832q^{2} + \)\(40\!\cdots\!28\)\(q^{3} + \)\(30\!\cdots\!56\)\(q^{4} - \)\(40\!\cdots\!60\)\(q^{5} - \)\(35\!\cdots\!24\)\(q^{6} + \)\(85\!\cdots\!04\)\(q^{7} - \)\(27\!\cdots\!48\)\(q^{8} + \)\(82\!\cdots\!48\)\(q^{9} + \)\(35\!\cdots\!80\)\(q^{10} + \)\(29\!\cdots\!48\)\(q^{11} + \)\(31\!\cdots\!92\)\(q^{12} - \)\(17\!\cdots\!12\)\(q^{13} - \)\(75\!\cdots\!32\)\(q^{14} + \)\(33\!\cdots\!80\)\(q^{15} + \)\(23\!\cdots\!84\)\(q^{16} - \)\(70\!\cdots\!16\)\(q^{17} - \)\(72\!\cdots\!84\)\(q^{18} + \)\(38\!\cdots\!60\)\(q^{19} - \)\(31\!\cdots\!40\)\(q^{20} - \)\(29\!\cdots\!72\)\(q^{21} - \)\(26\!\cdots\!84\)\(q^{22} + \)\(26\!\cdots\!28\)\(q^{23} - \)\(27\!\cdots\!36\)\(q^{24} + \)\(26\!\cdots\!00\)\(q^{25} + \)\(15\!\cdots\!96\)\(q^{26} + \)\(98\!\cdots\!00\)\(q^{27} + \)\(66\!\cdots\!56\)\(q^{28} - \)\(93\!\cdots\!40\)\(q^{29} - \)\(29\!\cdots\!40\)\(q^{30} + \)\(11\!\cdots\!68\)\(q^{31} - \)\(21\!\cdots\!72\)\(q^{32} - \)\(97\!\cdots\!64\)\(q^{33} + \)\(61\!\cdots\!28\)\(q^{34} + \)\(14\!\cdots\!40\)\(q^{35} + \)\(63\!\cdots\!72\)\(q^{36} + \)\(18\!\cdots\!64\)\(q^{37} - \)\(33\!\cdots\!80\)\(q^{38} - \)\(51\!\cdots\!84\)\(q^{39} + \)\(27\!\cdots\!20\)\(q^{40} + \)\(54\!\cdots\!08\)\(q^{41} + \)\(26\!\cdots\!76\)\(q^{42} - \)\(15\!\cdots\!52\)\(q^{43} + \)\(22\!\cdots\!72\)\(q^{44} - \)\(13\!\cdots\!20\)\(q^{45} - \)\(23\!\cdots\!24\)\(q^{46} - \)\(11\!\cdots\!36\)\(q^{47} + \)\(24\!\cdots\!88\)\(q^{48} + \)\(45\!\cdots\!32\)\(q^{49} - \)\(23\!\cdots\!00\)\(q^{50} - \)\(91\!\cdots\!12\)\(q^{51} - \)\(13\!\cdots\!68\)\(q^{52} - \)\(32\!\cdots\!12\)\(q^{53} - \)\(86\!\cdots\!00\)\(q^{54} - \)\(10\!\cdots\!20\)\(q^{55} - \)\(58\!\cdots\!48\)\(q^{56} + \)\(24\!\cdots\!20\)\(q^{57} + \)\(82\!\cdots\!20\)\(q^{58} - \)\(81\!\cdots\!80\)\(q^{59} + \)\(26\!\cdots\!20\)\(q^{60} + \)\(90\!\cdots\!28\)\(q^{61} - \)\(10\!\cdots\!44\)\(q^{62} + \)\(71\!\cdots\!48\)\(q^{63} + \)\(18\!\cdots\!76\)\(q^{64} + \)\(11\!\cdots\!80\)\(q^{65} + \)\(85\!\cdots\!12\)\(q^{66} - \)\(65\!\cdots\!96\)\(q^{67} - \)\(54\!\cdots\!24\)\(q^{68} - \)\(14\!\cdots\!04\)\(q^{69} - \)\(12\!\cdots\!20\)\(q^{70} - \)\(18\!\cdots\!92\)\(q^{71} - \)\(55\!\cdots\!76\)\(q^{72} + \)\(35\!\cdots\!88\)\(q^{73} - \)\(16\!\cdots\!12\)\(q^{74} + \)\(71\!\cdots\!00\)\(q^{75} + \)\(29\!\cdots\!40\)\(q^{76} + \)\(39\!\cdots\!48\)\(q^{77} + \)\(45\!\cdots\!72\)\(q^{78} + \)\(15\!\cdots\!20\)\(q^{79} - \)\(24\!\cdots\!60\)\(q^{80} - \)\(19\!\cdots\!76\)\(q^{81} - \)\(48\!\cdots\!64\)\(q^{82} + \)\(11\!\cdots\!68\)\(q^{83} - \)\(22\!\cdots\!08\)\(q^{84} - \)\(89\!\cdots\!60\)\(q^{85} + \)\(13\!\cdots\!16\)\(q^{86} + \)\(20\!\cdots\!20\)\(q^{87} - \)\(20\!\cdots\!76\)\(q^{88} - \)\(11\!\cdots\!60\)\(q^{89} + \)\(12\!\cdots\!60\)\(q^{90} + \)\(79\!\cdots\!88\)\(q^{91} + \)\(20\!\cdots\!92\)\(q^{92} + \)\(12\!\cdots\!76\)\(q^{93} + \)\(10\!\cdots\!88\)\(q^{94} + \)\(36\!\cdots\!00\)\(q^{95} - \)\(21\!\cdots\!04\)\(q^{96} + \)\(84\!\cdots\!84\)\(q^{97} - \)\(40\!\cdots\!56\)\(q^{98} + \)\(22\!\cdots\!76\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 281103089248735483936525960163617359 x^{2} - 12802046688892212650602531506830185803155982810145588 x + 13431538901013885043865389957038837902382364731827315409437340098095268\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 1920 \nu - 960 \)
\(\beta_{2}\)\(=\)\((\)\(-9433750962176 \nu^{3} + 1769549267668589556685668835328 \nu^{2} + 1629454524675798842203795893047621524173694499456 \nu - 158134392657869780369087954106442708349250693952206168207850810688\)\()/ \)\(52\!\cdots\!57\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-10114407221690368 \nu^{3} + 12622111923347229353644701057531904 \nu^{2} + 1014369494928421110790943986822141502075067313900288 \nu - 1676943492136190755850952090281228658060462045559125237479226421642624\)\()/ \)\(57\!\cdots\!27\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 960\)\()/1920\)
\(\nu^{2}\)\(=\)\((\)\(98963986 \beta_{3} - 9645850393 \beta_{2} + 65580735096668436823 \beta_{1} + 259064607051634621995902324886789759897600\)\()/1843200\)
\(\nu^{3}\)\(=\)\((\)\(2376103752972959609807416649 \beta_{3} - 1540785146892972333208581672062 \beta_{2} + 22799155987924903377416244937247733974012 \beta_{1} + 2265286315868748229809505370873077099774695078030660953702400\)\()/ 235929600 \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.03541e17
2.13103e17
−3.01536e17
−4.15109e17
−8.79609e12 −8.65990e20 7.73713e25 −4.55634e30 7.61733e33 8.09122e36 −6.80565e38 4.26681e41 4.00780e43
1.2 −8.79609e12 −3.08348e20 7.73713e25 1.64822e30 2.71226e33 −1.87890e36 −6.80565e38 −2.28179e41 −1.44979e43
1.3 −8.79609e12 6.79758e20 7.73713e25 −4.34651e30 −5.97921e33 −6.14091e36 −6.80565e38 1.38813e41 3.82323e43
1.4 −8.79609e12 8.97819e20 7.73713e25 3.21672e30 −7.89730e33 8.51076e36 −6.80565e38 4.82821e41 −2.82946e43
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.88.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.88.a.b 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - \)\(40\!\cdots\!28\)\( T_{3}^{3} - \)\(97\!\cdots\!56\)\( T_{3}^{2} + \)\(29\!\cdots\!28\)\( T_{3} + \)\(16\!\cdots\!76\)\( \) acting on \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8796093022208 T )^{4} \)
$3$ \( 1 - \)\(40\!\cdots\!28\)\( T + \)\(31\!\cdots\!92\)\( T^{2} - \)\(95\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} - \)\(30\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!48\)\( T^{6} - \)\(13\!\cdots\!84\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$5$ \( 1 + \)\(40\!\cdots\!60\)\( T + \)\(76\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(85\!\cdots\!04\)\( T + \)\(80\!\cdots\!28\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(90\!\cdots\!72\)\( T^{6} - \)\(31\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(29\!\cdots\!48\)\( T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(30\!\cdots\!36\)\( T^{3} + \)\(82\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!28\)\( T^{7} + \)\(25\!\cdots\!81\)\( T^{8} \)
$13$ \( 1 + \)\(17\!\cdots\!12\)\( T + \)\(97\!\cdots\!72\)\( T^{2} + \)\(46\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!40\)\( T^{5} + \)\(65\!\cdots\!08\)\( T^{6} + \)\(96\!\cdots\!56\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 + \)\(70\!\cdots\!16\)\( T + \)\(43\!\cdots\!88\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} + \)\(75\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!52\)\( T^{6} + \)\(98\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \)
$19$ \( 1 - \)\(38\!\cdots\!60\)\( T + \)\(51\!\cdots\!56\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!76\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 - \)\(26\!\cdots\!28\)\( T + \)\(13\!\cdots\!32\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!66\)\( T^{4} - \)\(70\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!88\)\( T^{6} - \)\(68\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 + \)\(93\!\cdots\!40\)\( T + \)\(53\!\cdots\!36\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(56\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(45\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 - \)\(11\!\cdots\!68\)\( T + \)\(10\!\cdots\!28\)\( T^{2} + \)\(10\!\cdots\!04\)\( T^{3} - \)\(17\!\cdots\!30\)\( T^{4} + \)\(61\!\cdots\!44\)\( T^{5} + \)\(33\!\cdots\!88\)\( T^{6} - \)\(21\!\cdots\!08\)\( T^{7} + \)\(98\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 - \)\(18\!\cdots\!64\)\( T + \)\(10\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!46\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{5} + \)\(75\!\cdots\!52\)\( T^{6} - \)\(36\!\cdots\!68\)\( T^{7} + \)\(54\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 - \)\(54\!\cdots\!08\)\( T + \)\(31\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(71\!\cdots\!70\)\( T^{4} - \)\(31\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(47\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + \)\(15\!\cdots\!52\)\( T + \)\(51\!\cdots\!92\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!26\)\( T^{4} + \)\(70\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!08\)\( T^{6} + \)\(34\!\cdots\!36\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + \)\(11\!\cdots\!36\)\( T + \)\(12\!\cdots\!88\)\( T^{2} + \)\(81\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!92\)\( T^{7} + \)\(77\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 + \)\(32\!\cdots\!12\)\( T + \)\(67\!\cdots\!52\)\( T^{2} + \)\(96\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} + \)\(98\!\cdots\!80\)\( T^{5} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(34\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 + \)\(81\!\cdots\!80\)\( T + \)\(31\!\cdots\!76\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(42\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 - \)\(90\!\cdots\!28\)\( T + \)\(94\!\cdots\!28\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(41\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!08\)\( T^{7} + \)\(19\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 + \)\(65\!\cdots\!96\)\( T + \)\(27\!\cdots\!48\)\( T^{2} + \)\(80\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!26\)\( T^{4} + \)\(59\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 + \)\(18\!\cdots\!92\)\( T + \)\(37\!\cdots\!88\)\( T^{2} + \)\(47\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!70\)\( T^{4} + \)\(54\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!32\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 - \)\(35\!\cdots\!88\)\( T + \)\(28\!\cdots\!92\)\( T^{2} - \)\(92\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(47\!\cdots\!28\)\( T^{6} - \)\(75\!\cdots\!24\)\( T^{7} + \)\(27\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 - \)\(15\!\cdots\!20\)\( T + \)\(22\!\cdots\!36\)\( T^{2} - \)\(33\!\cdots\!40\)\( T^{3} + \)\(41\!\cdots\!86\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} - \)\(29\!\cdots\!80\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 - \)\(11\!\cdots\!68\)\( T + \)\(23\!\cdots\!92\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!06\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(84\!\cdots\!44\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(10\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!46\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!56\)\( T^{6} + \)\(68\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 - \)\(84\!\cdots\!84\)\( T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(97\!\cdots\!66\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(70\!\cdots\!12\)\( T^{6} - \)\(29\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!61\)\( T^{8} \)
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